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\noindent{\bf Math 431 homework}\hfill Due Friday, October 26, 2001

\ex
Let $I = (2,1+\sqrt{-3}) \subset \Z[\sqrt{-3}]$.  Show
that $I^2 = 2I$.  Why does this not contradict unique factorization
of ideals in a Dedekind domain?

\ex
Let $F = \Q(\alpha)$, where $\alpha^3-\alpha^2-2\alpha-8 = 0$.
Show that $\Z_F = \spn_\Z(1,\alpha,(\alpha^2+\alpha)/2)$.
Find three nontrivial ring homomorphisms from $\Z_F$ to $\F_2$.

\ex
Show that any Dedekind domain with finitely many prime ideals is a
PID.  (Hint: Use the CRT to find a generator of an ideal.)

\ex
Show that if $I$ is any ideal in a number ring $\Z_F$ then
$\Z_F/I$ is a PID.

\ex
Let $F = \Q(\alpha)$ where $\alpha^3+\alpha+1 = 0$.
Show that in the ring $\Z_F$
$$(3) = PQ, \mbox{ where } P = (3,\alpha+2), Q = (3,\alpha^2+\alpha+2)$$
$$(31) = P^2Q, \mbox{ where } P = (31,\alpha-14), Q = (31,\alpha-3).$$

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